Chern-Simons functional, singular instantons, and the four-dimensional clasp number

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Chern-Simons functional, singular instantons, and the

four-dimensional clasp number

Aliakbar Daemi∗ Christopher Scaduto†

Abstract

Kronheimer and Mrowka asked whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large. This question is answered affirmatively by studying a knot invariant derived from equivariant singular instanton theory, and which is closely related to the Chern–Simons functional. This also an-swers a conjecture of Livingston about slicing numbers. Also studied is the singular instanton Frøyshov invariant of a knot. If defined with integer coefficients, this gives a lower bound for the unoriented slice genus, and is computed for quasi-alternating and torus knots. In contrast, for certain other coefficient rings, the invariant is identified with a multiple of the knot signature. This result is used to address a conjecture by Poudel and Saveliev about traceless SU p2q representations of torus knots. Further, for a concordance between knots with non-zero signature, it is shown that there is a traceless representation of the concordance complement which restricts to non-trivial representations of the knot groups. Finally, some evidence towards an extension of the slice-ribbon conjecture to torus knots is provided.

∗

The work of AD was supported by NSF Grant DMS-1812033 and NSF FRG Grant DMS-1952762. †

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1 Introduction

In previous work [DS], the authors introduced a framework for studying S1-equivariant instanton Floer theoretic invariants for a knot K in an integer homology 3-sphere Y . Morally the constructions are derived from Morse theory for the Chern–Simons functional on the space of singular SU p2q connections framed at a basepoint on K. This space has an S1 -action, and the critical set is acted on freely except for a unique isolated fixed point. This critical set is naturally in bijection with traceless representations of π1pY zKq into SU p2q, i.e., SU p2q-representations of the knot group which send a distinguished meridian near the basepoint of K to i P SU p2q.

The details of the constructions rely on the extensive work of Kronheimer and Mrowka, who developed the foundations of singular instanton Floer theory [KM11b,KM11a]. We show in [DS] that many of the invariants defined by Kronheimer and Mrowka can be recovered from the equivariant framework.

In this paper we continue our study of the theory initiated in [DS], with an emphasis on topological applications. We carry out computations for several families of 2-bridge knots and torus knots. Using the structure of the Chern–Simons filtration and computations of singular Frøyshov invariants, we obtain information about the four-dimensional clasp number of knots, fundamental groups of concordance complements, and more.

The four-dimensional clasp number of a knot

A normally immersed surface is a smoothly immersed surface in a manifold which has transverse double points. The 4-dimensional clasp number cspKq of a knot K is the minimal number of double points realized by a normally immersed disk in B4 with boundary K [Shi74]. In the literature cspKq is also called the 4-ball crossing number [OS16]. Variations of cscan be defined by keeping track of the signs of double points. Let c`spKq (resp. c´spKq) denote the minimal number of positive (resp. negative) double points realized by a normally immersed disk in B4 with boundary K. Then cspKq ě c`spKq ` c´spKq. Recall that two knots K and K1 _{in the 3-sphere are concordant if they cobound a properly smoothly} embedded annulus in r0, 1s ˆ S3. Clearly cspKq and c˘spKq are concordance invariants.

A standard construction allows us to remove a double point of a normally immersed surface at the expense of increasing the genus. We thus have

cspKq ě gspKq

where gspKq is the slice genus of K, the minimal genus of a smoothly embedded oriented surface in the 4-ball which has boundary K. Kronheimer and Mrowka asked [KM,Kro]: Question 1. Can the difference cspKq ´ gspKq be arbitrarily large?

A more refined question is the following which is already implicit in [KM19b]. Question 2. Can the difference c`

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Part of the subtlety of Question 2 is that many of the known knot invariants which provide lower bounds for gspKq do not give better bounds for c`spKq. For example, the signature σ of a knot K satisfies

|1 2σpKq| ď gspKq, ´ 1 2σpKq ď c ` spKq, 1 2σpKq ď c ´ spKq. (1.1) Similar inequalities hold for Levine-Tristram signatures σwpKq, Heegaard Floer invariants τ pKq and ΥpKq [OS03,Ras03a,OSS17a,JZ20] and Rasmussen’s invariant spKq [Ras10,

MMSW19]. In [DS], the authors defined various concordance invariants. In this paper, we study the behavior of these invariants with respect to knot cobordisms and use them to prove the following theorem.

Theorem 1. Let K1 be the knot74, which is the p15, 4q 2-bridge knot, and let Knbe the n-fold connected sum of K1. Then we have:

c`

spKnq ´ gspKnq ě n{5.

As a corollary of this theorem, one obtains a positive answer to Question 1 by considering n-fold connected sums of 74. Although σwpKq, τ pKq, ΥpKq and spKq cannot be used to answer Question 2 directly, they can be useful to answer Question 1. The idea is to use the analogue of the second inequality in (1.1) for one of these invariants to provide a lower bound for c`

spKq and the analogue of the third inequality for another invariant to give a lower bound for c´

spKq. Then combining these lower bounds gives rise to a lower bound for cspKq which potentially could be as large as 2gspKq. In a recent article [JZ20], Juhász and Zemke answered Question 1 independently by applying this strategy using ΥpKq where K is given by connected sums of certain torus knots. More recently, Feller and Park showed that Levine-Tristram signatures can be also used to answer Question 1 in the case that K is again given by connected sums of torus knots [FP20].

Theorem 1 also addresses a conjecture made in [Liv02]. The slicing number of a knot K is the minimum number of crossing changes required to convert K into a slice knot. Any sequence of i crossing changes, which slices K, can be used to form an immersed disk in B4 with boundary K and i double points. Thus the slicing number of K is at least as large as cspKq. A refinement keeps track of positive and negative crossing changes. Let I Ă Z2ě0be the set of pairs pm, nq such that a collection of m positive crossing changes and n negative crossing changes converts K into a slice knot. Livingston defines

UspKq :“ min pm,nqPI

maxpm, nq.

Then Usis again bounded below by gsand Livingston asked if Us´ gscan be arbitrarily large [Liv02, Conjecture 5.4]. Since c`

spKq ď UspKq, we have the following corollary. Corollary 1. The quantity UspKnq ´ gspKnq is at least n{5, and hence the difference Us´ gscan be arbitrarily large.

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It is easy to construct a genus 1 slicing surface for K1(see Figure 1) and thus a genus n slicing surface for Kn. Murasugi’s inequality gspKq ě |1₂σpKq| and the computation σpKnq{2 “ ´n imply gspKnq “ n. So the non-trivial part of Theorem 1 is to provide the lower bound 6n{5 for c`

spKnq. The main result of [Liv02] states that the slicing number of K1is 2 and c`spK1q “ 2 is computed in [OS16]. It is suggested in [KM] that the knots Knmay give a family for which cs´ gsbecomes arbitrarily large. The tools developed in [KM19b] by Kronheimer and Mrowka could potentially detect a sequence of knots with cs´ gs being arbitrarily large. However, the particular invariants of that article do not reproduce the above result for n74.

We outline the proof of Theorem 1. The main tool is the concordance invariant ΓKfor a knot K, constructed in [DS]. It is derived from the structure of the Chern–Simons filtration on the equivariant singular instanton complex for K, and is an analogue of the invariant ΓY for homology 3-spheres defined in [Dae18]. The invariant ΓK is a function from the integers with values in Rě0Y 8. We will show:

Theorem 2. For K Ă S3, if ´σpKq{2 ě 0, then we have the inequality: ΓK ˆ ´1 2σpKq ˙ ď 1 2c ` spKq.

Proof of Theorem 1. Let Kn“ n74. In Proposition 3.28 we compute:

ΓKnpiq “ $ ’ & ’ % 0, i ď 0 3 5i, 1 ď i ď n 8, i ě n ` 1 Theorem 2 implies ΓKnpnq “ 3n{5 ď c ` spKnq{2, and so c`spKnq ě 6n{5.

We can prove similar results for other families of knots, such as connected sums of double twist knots Dm,n. The double twist knot Dm,nis given in Figure 1. It is a 2-bridge knot with gspDm,nq “ 1 “ ´σpDm,nq{2 and in fact D2,2 “ 74.

Theorem 3. For m, n P Zą0letDm,n be the double twist knot. Fork P Zą0we have: c`_spkDm,nq ´ gspkDm,nq ě

kp4mn ´ 4m ´ 4n ` 3q

4mn ´ 1 .

Thus ifm, n ě 2 then c`

spkDm,nq ´ gspkDm,nq ě Ck for some C ą 0.

Recall that the unknotting number upKq is the minimal number of crossing changes needed to turn K into the unknot U1. Performing these crossing changes in reverse order, to get from U1 to K, induces a movie in S3ˆ r0, 1s, which is an immersed annulus with transverse double points. Capping off the unknot component of our annulus gives a normally immersed disk in B4with boundary K, with upKq double points. Thus upKq ě c4pKq ě c`

spKq, and Theorem 2 also gives a lower bound on upKq. We can use this to determine the unknotting numbers of some 11-crossing 2-bridge knots.

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Theorem 4. The knots 11a192, 11a341, 11a360 each have unknotting number equal to 3. The unknotting number of11a365is equal to4. In fact, for each of these knots, c`s and the 4-dimensional clasp number are equal to the unknotting number.

The unknotting number of 11a365 was previously computed by Owens [Owe08], and its 4-dimensional clasp number follows from the work of Owens and Strle [OS16]. The other three examples in the theorem are new, as far as the authors are aware.

Remark1.2. Theorem 1 addresses a relative version of the following problem. Let W be a smooth closed 4-manifold and σ P H2pW ; Zq. Let gpσq be the minimal genus over all connected, orientable smoothly embedded representatives of σ, and c`

pσq be the minimal number of positive double points over normally immersed sphere representatives. How large can c`_{pσq ´ gpσq be? This question has been advertised by Kronheimer and Mrowka,} partly motivated by its relationship to the Caporaso-Harris-Mazur conjecture about algebraic curves in a quintic surface [Kro].

Signatures, Frøyshov invariants, and representations

For a knot K in the 3-sphere, the knot group π1pS3zKq is infinite cyclic if and only if K is an unknot, as follows from the work of Papakyriakopoulos [Pap57]. Using instanton gauge theory, Kronheimer and Mrowka showed that in fact for a non-trivial knot there exists a non-abelian representation from π1pS3zKq into SU p2q [KM04].

There are analogous problems in 4-dimensions. Given a 2-knot, i.e. a smoothly em-bedded 2-sphere F in S4, if π1pS4zF q is infinite cyclic, is F an unknotted 2-sphere? If we instead work in the topologically locally flat category, then Freedman’s work [Fre84] answers this in the affirmative. In either category, we may further ask: given a non-trivial 2-knot F , does π1pS4zF q admit a non-abelian representation to SU p2q?

We may also pose analogous questions about embedded annuli in r0, 1s ˆ S3. The following gives a simple criterion for the existence of non-abelian SU p2q representations in this scenario under an assumption involving the knot signature.

Theorem 5. If S Ă r0, 1s ˆ S3 is a smooth concordance from a knot K to itself, and σpKq ‰ 0, then π1pr0, 1s ˆ S3zSq admits a non-abelian representation to SU p2q.

The above generalizes as follows. A homology concordance is a cobordism of pairs pW, Sq : pY, Kq Ñ pY1, K1

q between integer homology 3-spheres with knots such that W is an integer homology cobordism and S is a properly embedded annulus. The collection of integer homology 3-spheres with knots modulo homology concordance gives rise to an abelian group CZ. There is a homomorphism from the smooth concordance group to C_Z. There is also a homomorphism Θ3_Z Ñ CZ where Θ3Z is the homology cobordism group of integer homology 3-spheres, induced by Y ÞÑ pY, unknotq. Write hpY q P Z for the instanton Frøyshov invariant for the integer homology 3-sphere Y , which is a homomorphism h : Θ3

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Theorem 6. Suppose pW, Sq : pY, Kq Ñ pY1_{, K}1

q is a homology concordance where K is null-homotopic in the integer homology 3-sphereY . Suppose further that:

´1

2σpY, Kq ` 4hpY q ‰ 0. (1.3)

Then there exists a traceless representationπ1pW zSq Ñ SU p2q which extends non-abelian traceless representations ofπ1pY zKq and π1pY1zK1q.

Here “traceless” means that the representation sends classes of the circle fibers of the unit circle normal bundle of S (resp. K, K1_{) to traceless elements of SU p2q. Theorem 6} should be compared to analogous results in the context of homology cobordism of homology 3-spheres (without knots), see [Tau87,Dae18].

The authors established in [DS, Theorem 1.17] a version of Theorem 6 with the left hand side of (1.3) replaced by h_SpY, Kq, the singular instanton Frøyshov invariant. This invariant, also introduced in [DS], is a homomorphism h_SpY, Kq : CZ Ñ Z, and is defined for a coefficient ringS which is an integral domain algebra over ZrU˘1_{, T}˘1_{s. Thus Theorem 6} follows from our previous work and the following result, which identifies h_SpY, Kq with previously known invariants under some assumptions. We write h_SpS3, Kq “ h_SpKq. Theorem 7. LetS “ ZrU˘1_{, T}˘1

s. If K is a knot in the 3-sphere, then we have: h_SpKq “ ´1

2σpKq. (1.4)

IfK is a null-homotopic knot in an integer homology 3-sphere Y , then: h_SpY, Kq “ ´1

2σpY, Kq ` 4hpY q. (1.5)

The result(1.4) holds more generally whenS is an integral domain algebra over ZrU˘1_{, T}˘1_s withT4 ‰ 1. Further, (1.5) holds more generally as well; see Section 5.

The identification of h_SpKq in terms of the classical knot signature helps elucidate the general structure of the singular instanton Floer invariants of K, as we now explain. Let I˚pY, Kq denote the irreducible singular instanton homology of pY, Kq, defined in [DS]. It is a Z{4-graded abelian group and its euler characteristic satisfies

χ pI˚pY, Kqq “ 1

2σpY, Kq ` 4λpY q

where λpY q is the Casson invariant of Y , a result essentially due to Herald [Her97]. More generally, we can define anS -module I˚pY, K; ∆Sq, where as beforeS is an algebra over ZrU˘1_{, T}˘1_{s, and ∆}

S is a local coefficient system overS .

Corollary 2. Suppose T4 ‰ 1 inS . If K Ă S3andσpKq ď 0 then we have inequalities: rkpI1pK; ∆Sqq ě R ´σpKq 4 V , rkpI3pK; ∆Sqq ě Z ´σpKq 4 ^ . (1.6)

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Inequalities in the case σpKq ą 0 are obtained using rkpIip´K; ∆Sqq “ rkpI3´ipK; ∆Sqq and σp´Kq “ ´σpKq. Corollary 2 follows from Theorem 7 and elementary inequalities relating the irreducible instanton homology to the Frøyshov invariant, see (2.6). See Theorem 10 for a stronger version of Corollary 2.

The following was essentially conjectured in [PS17, Section 7.4].

Corollary 3. Let Tp,qbe the pp, qq torus knot. Then as a Z{4-graded abelian group, I˚pTp,qq – Zr´σpT_p1q p,qq{4s‘ Zt´σpT_p3q p,qq{4u. (1.7) Proof. The result in fact holds for any coefficient ringS . From [PS17] and [DS, Section 9.5], the chain complex for the irreducible homology of Tp,q has total rank ´σpTp,qq{2 and is supported in gradings 1 and 3 (mod 4). The differential is necessarily zero. The lower bounds from Corollary 2 imply the result when T4 ‰ 1 inS , and the ranks of the Z{4-graded chain complex are independent of the ringS .

Another strategy to prove Corollary 3 involves directly computing the mod 4 gradings of the singular flat connections. Anvari computed these mod 4 gradings in terms of cer-tain arithmetic functions and verified Corollary 3 for the p3, 6k ` 1q torus knots [Anv19]. Corollary 3 has implications about the arithmetic functions appearing in [Anv19].

The special structure of the instanton homology for torus knots described above will be used to upgrade Theorem 6 in the case that one of the knots is a torus knot. For simplicity we state the result for classical concordances in r0, 1s ˆ S3.

Theorem 8. Suppose S : Tp,q Ñ K is a smooth concordance. Then every traceless SU p2q representation ofπ1pS3zTp,qq extends over the concordance complement.

A consequence of this result is that if a knot K is concordant to a torus knot Tp,qthen the (singular) Chern–Simons invariants of Tp,qare a subset of those of K. We also have the following relation at the level of Floer homology.

Theorem 9. Suppose K is a knot which is concordant to a torus knot Tp,q. ThenI˚pTp,qq is a module summand ofI˚pKq for any choice of ordinary or local coefficient system. The same holds for other versions of singular instanton homology, such asI˚6pKq and pI˚pKq.

This type of result for instanton Floer homology also holds when two knots are related by a ribbon concordance, see e.g. [DLSVVW20], [KM19b, Theorem 7.4]. This motivates the following, which is closely related to the slice-ribbon conjecture.

Question 3. If there is a concordance from a torus knot Tp,qto another knotK, is there then a ribbon concordance fromTp,qtoK?

Our convention is that a ribbon concordance from a knot K0 to a knot K1 is a properly embedded surface S in r0, 1s ˆ S3 such that S X tiu ˆ S3 “ Kiand the projection of S to r0, 1s is a Morse function without local maxima. (In particular, S is a ribbon concordance from K1to K0in the convention of [Gor81].)

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Turning back to the situation of more general knots in S3, we now describe an algebraic refinement of Theorem 7. The equivariant singular instanton theory developed in [DS] associates to a knot K a Z{4-graded chain complex rC˚pK; ∆Sq overS . This complex has the additional structure of an S-complex, which formalizes the homological properties of the S1-action on the framed configuration space. The homotopy type of the S-complex

r

C˚pK; ∆Sq is an invariant of K, and determines the invariant hSpKq. Thus the following result is an upgrade of the computation h_SpKq “ ´σpKq{2 from Theorem 7.

Theorem 10. If T4 ´ 1 is invertible inS , then the S-complex rC˚pK; ∆Sq is homotopy equivalent to rC˚pεT2,2k`1; ∆Sq, where k “ |σpKq|{2 and ε is the sign of ´σpKq. As a consequence, in the inequalities(1.6) of Corollary 2, equality holds.

Furthermore, the S-complex for T2,2k`1 is very simple. We prove a similar statement for any null-homotopic knot in an integer homology 3-sphere. See Theorem 5.9.

Theorem 10 shows that if our coefficient ringS has T4 ‰ 1, then the ranks of the irreducible instanton homology groups are determined by the signature of the knot in the same way as for torus knots in (1.7). In the special case of a 2-bridge knot, it was shown in [DS, Theorem 1.12] that the irreducible instanton homology for a certain local coefficient system with T4 ‰ 1 determines a version of lens space instanton homology, denoted I˚pLpp, qqq, defined by Furuta [Fur90] and Sasahira [Sas13]. The latter is a Z{4-graded vector space over F, the field with 2 elements. In [Sas13], Sasahira computed these groups for the family Lp8N ` 1, 2q. Combining [DS, Theorem 1.12] with Theorem 10 we obtain: Corollary 4. Suppose σpKp,qq ď 0 where Kp,qis the 2-bridge knot with double branched coverLpp, ´qq. Then the lens space instanton homology of Furuta and Sasahira is given by

I˚pLpp, ´qqq – F

r´σpKp,qq{4s

p1q ‘ F

t´σpKp,qq{4u p3q

as aZ{4-graded F-vector space. If σpKp,qq ą 0, use I˚pLpp, ´qqq – I3´˚pLpp, qqq. In [Sas13], Sasahira proved a gluing result for 2-torsion instanton invariants of 4-manifolds which decompose along lens spaces. Using this result and knowledge of I˚pLpp, qqq, he studied the question of whether CP2#CP2can be decomposed along Lpp, qq.

Instanton knot ideals

In recent work [KM19b], Kronheimer and Mrowka introduced a concordance invariant for a knot K Ă S3 from singular instanton theory which takes the form of a fractional ideal z6_BNpKq over the ring FrT1˘1, T2˘2, T3˘1s, where F is the field with two elements. Recall that a fractional ideal I over an integral domain R is an R-submodule in the field of fractions of R such that there is some r P R so that r ¨ I is an ideal in R.

Fix an integral domain algebraS over ZrU˘1_{, T}˘1

s. The definition of z_BN6 pKq was adapted in [DS] to define a fractional idealpzpKq over the ringS rxs for knots K satisfy-ing gspKq “ ´σpKq{2, in the framework of equivariant singular instanton theory. The restriction here on the class of knots is due to the limited functoriality developed in [DS].

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Using the idea of splicing or “suspending” an arbitrary cobordism by some elementary cobordisms involving the trefoil, we extend the construction ofpzpKq to all knots K, such that the relationship to z6_BNpKq established in [DS, Section 8.8] continues to hold.

Theorem 11. To a knot K we construct a fractional ideal pzpKq overS rxs which is a concordance invariant. IfS “ FrT˘1

s, then z6_BNpKq can be recovered frompzpKq as z_BN6 pKq “zpKq bp S rxsFrT ˘1 1 , T ˘2 2 , T ˘3 3 s

where theS rxs-module structure of FrT₁˘1, T₂˘2, T₃˘3s is determined by T ÞÑ T1and x ÞÑ T1T2T3` T1T2´1T3´1` T1´1T2T3´1` T1´1T2´1T3.

The idealpzpKq has the advantage over z 6

BNpKq that it is not restricted to characteristic 2 rings. Further, for the choiceS “ FrT˘1

s, the idealpzpKq is a fractional ideal over the ring FrT˘1_{, xs, instead of over the more complicated ring FrT}˘1

1 , T ˘2 2 , T

˘3

3 s. Thus Theorem 11 also restricts the types of ideals realized by z_BN6 pKq.

The singular Frøyshov invariant and the unoriented 4-ball genus

The result of Theorem 7 shows that the singular instanton Frøyshov invariant for a knot in the 3-sphere is determined by the signature, so long as we work with a coefficient ringS for which T4 ‰ 1. When T4 “ 1 the situation is rather different, as already evidenced by the computations in [DS]. The following highlights this dichotomy further.

Proposition 1. Suppose T4 “ 1 inS . Then we have the following computations for h_S: (i) If K is alternating, or more generally quasi-alternating, then h_SpKq “ 0.

(ii) For torus knots, h_S is determined recursively by: for coprimep, q P Zą0, h_SpTp,p`qq ` 1 2σpTp,p`qq “ hSpTp,qq ` 1 2σpTp,qq ´Xp 2 {4\ .

These computations show that in the case T4 “ 1, for quasi-alternating knots and torus knots the invariant h_SpKq ` 1₂σpKq agrees with the Heegaard Floer invariant υpKq (upsilon of K) defined by Ozsváth–Stipsicz–Szabó [OSS17b] and also with the invariant ´1₂tpKq defined by Ballinger [Bal20] in the setting of Khovanov homology.

An important ingredient in the computations of Proposition 1 is the relationship between h_SpKq and the unoriented 4-ball genus γ4pKq. Also called the 4-dimensional crosscap number, γ4pKq is defined to be the minimal b1pSq “ b1pS; Z{2q among all smoothly embedded possibly non-orientable surfaces S in the 4-ball with boundary K. Clearly γ4pKq ď 2gspKq. We have the following:

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Theorem 12. Suppose T4“ 1 inS . Suppose S is a properly smoothly embedded (possibly non-orientable) surface in the 4-ball with boundary the knotK. Then

ˇ ˇ ˇ ˇ h_SpKq ` 1 2σpKq ´ 1 4S ¨ S ˇ ˇ ˇ ˇď b1pSq. (1.8)

The term S ¨ S should be interpreted as the euler number of the normal bundle of S. Just as in [Bat14,OSS17b,Bal20], we may combine the inequality (1.8) with an inequality |σpKq ´1₂S ¨ S| ď b1pSq of Gordon–Litherland [GL78a] to obtain the following.

Theorem 13. Suppose T4 “ 1 inS . For a knot K Ă S3we have |hSpKq| ď γ4pKq.

These inequalities are also satisfied upon replacing h_SpKq ` 1₂σpKq by the invariant υpKq [OSS17b, Theorems 1.1, 1.2] and also by ´1₂tpKq [Bal20, Theorem 1.1]. We remark that Theorem 13 applied to connected sums of T3,4shows singular instanton theory detects that γ4pKq can be arbitrarily large, a result first proved by Batson [Bat14, Theorem 2], and which can also be proved using the invariants υpKq and tpKq.

We will upgrade Proposition 1 (i) and show that in the case T4 “ 1, if K is a quasi-alternating knot then the associated Γ-invariant is the same as that for the unknot; see Proposition 7.10. We also remark that Theorem 6 holds with ´σpKq{2 replaced by h_SpKq when T4 “ 1 inS . Because in this situation h_S and σ are linearly independent concordance homomorphisms, this provides more examples of existence results for traceless SU p2q representations; see Theorem 7.15.

In [KM19b], Kronheimer and Mrowka define homomorphisms fσ from the smooth concordance group to Z for suitable base changes σ : FrT₀˘1, T˘1

1 ,˘1, T2˘1, T3˘1s Ñ S whereS is a ring and F is the field with 2 elements. If σpT0q “ σpT1q “ 1, then the corresponding invariant fσ behaves similarly to hS (here T4 “ 1), in that it satisfies the same kinds of inequalities; see Proposition 5.5 of [KM19b] and the discussion thereafter. It would be interesting to determine the precise relationship between these invariants.

Outline

Section 2 provides background on the equivariant singular instanton constructions from [DS]. The discussion therein of cobordism-induced morphisms is slightly more general than that of [DS], and is generalized even further in Section 4. In Section 2 we also consider cobordisms with immersed surfaces, and the relations for some basic cobordism moves in this setting are discussed following [Kro97]. Section 3 contains computations for 2-bridge knots, and we prove Theorems 3 and 4. In Section 4 we prove inequalities for singular Frøyshov invariants and for the ΓK invariant, of which Theorem 2 is a special case, and this in turn completes the proof of Theorem 1. Theorems 8 and 9 are also proved in Section 4. In Section 5 we prove Theorems 7 and 10. In Section 6, we construct concordance invariants in the form of ideals, and in particular prove Theorem 11. In Section 7, we discuss the invariant h_S in the case that T4“ 1 and prove Proposition 1 and Theorem 12.

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Acknowledgments

The authors thank Peter Kronheimer and Tom Mrowka for steering attention to the problem answered by Theorem 1, and for suggesting the potential utility of using connected sums of 74, and more generally double twist knots. The authors also thank Charles Livingston and Brendan Owens for helpful correspondences.

Conventions

We writeS for a commutative integral domain algebra over ZrU˘1_{, T}˘1_{s with unit. The} pair pY, Kq always denotes a knot K embedded in an integer homology 3-sphere Y . In the case that Y “ S3, we drop S3 from our notation whenever it does not cause any confusion. The unknot in the 3-sphere is denoted U1. If K is a knot in the 3-sphere, ´K denotes its mirror. Given a knot K and a positive integer n, we write nK for the n-fold connected sum of K. If n is negative, nK “ np´Kq, and 0K “ U1.

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2 Preliminaries

In this section we provide some background material. Most but not all of the material discussed is from [DS], and this section serves in part as an introduction to that more detailed reference. In the first two subsections, we review the algebra of S-complexes and their Frøyshov invariants, and briefly recall some of the invariants defined for knots from [DS]. Subsections 2.3 and 2.4 slightly generalize the class of cobordism-induced morphisms from [DS]. In Subsection 2.5 we review some relations on morphisms induced by cobordism moves following [Kro97]. In Subsection 2.6 we review the structure of the Chern–Simons filtration on the S-complex for a knot, and the resulting concordance invariants ΓpY,Kqare discussed in Subsection 2.7.

2.1 S-complexes and Frøyshov invariants

Let R be an integral domain. An S-complex over R is the data p rC˚, rd, χq where • p rC˚, rdq is a chain complex with rC˚a free R-module;

• χ is a degree 1 endomorphism of rC˚such that χ2 “ 0 and rdχ ` χ rd “ 0; • kerpχq{impχq is isomorphic to Rp0q, a copy of R in grading 0.

All S-complexes in the sequel will be Z{4-graded. A typical S-complex is given by r

C˚“ C˚‘ C˚´1‘ R where pC˚, dq is a chain complex and:

r d “ » – d 0 0 v ´d δ2 δ1 0 0 fi fl, χ “ » – 0 0 0 1 0 0 0 0 0 fi fl.

Here v : C˚ Ñ C˚´2, δ1 : C1 Ñ R and δ2 : R Ñ C´2 are maps that necessarily satisfy δ1d “ 0, dδ2“ 0, and dv ´ vd ´ δ2δ1 “ 0. In fact every S-complex is isomorphic to such an S-complex, and we therefore freely use this latter description.

A morphism rλ : rC˚ Ñ rC˚1 between two S-complexes rC˚ “ C˚ ‘ C˚´1 ‘ R and r

C1

˚“ C˚1 ‘ C˚´11 ‘ R is a chain map which has the form

r λ “ » – λ 0 0 µ λ ∆2 ∆1 0 η fi fl (2.1)

where η P R is a non-zero element. Here we slightly diverge from [DS], where a morphism was required to have η “ 1. However, all of the constructions easily extend to the case in which η is non-zero.

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An S-chain homotopy rC˚ Ñ rC˚1 between two morphisms is a chain homotopy in the usual sense which with respect to the decompositions has the form

» – K 0 0 L ´K M2 M1 0 0 fi fl.

A chain homotopy equivalence between S-complexes rC˚ and rC˚1 is a pair of morphisms r

λ : rC˚Ñ rC˚1 and rλ1 : rC˚1 Ñ rC˚such that rλ1rλ and rλrλ1 are S-chain homotopy equivalent to the identity morphisms.

Given S-complexes p rC˚, rd, χq and p rC˚1, rd1, χ1q we have the tensor product S-complex p rCb

˚, rdb, χbq defined by setting rC˚b“ rC˚b rC˚1 with the tensor product grading, and

r

db“ rd b 1 ` ε b rd1, χb “ χ b 1 ` ε b χ1. (2.3) Here ε is the “sign map” on rC1

˚, multiplying elements of homogeneous even (resp. odd) degree by `1 (resp. ´1). The dual of an S-complex p rC˚, rd, χq has underlying complex the dual chain complex, and with the dual of χ acting on it.

We say two S-complexes rC˚and rC˚1 are locally equivalent if there exist morphisms in both directions: rC˚ Ñ rC˚1 and rC˚1 Ñ rC˚. The equivalence classes form an abelian group ΘS_Rwith addition and inversion induced by tensor product and dual, respectively. We call ΘS_Rthe local equivalence group of S-complexes (over R and Z{4-graded).

The Frøyshov invariant is a surjective homomorphism h : ΘS_RÑ Z. It is characterized as follows, which the reader may take as a definition:

Proposition 2.4 (Prop. 4.15 of [DS]). The invariant hp rC˚q is positive if and only if there is anα P C1such thatdpαq “ 0 and δ1pαq ‰ 0. If hp rC˚q “ k for a positive integer k, then k is the largest integer such that there existsα P C˚satisfying the following properties:

dα “ 0, δ1vk´1pαq ‰ 0, δ1vipαq “ 0 fori ď k ´ 2.

If R is a field, then h : ΘS_RÑ Z is an isomorphism. In general, h : ΘS_RÑ Z factors through the isomorphism given by the Frøyshov invariant for the field of fractions of R.

2.2 Invariants for knots in homology spheres

Let pY, Kq be a pair of an oriented integer homology 3-sphere Y with a knot K Ă Y . In [DS], the authors associated to pY, Kq a Z{4-graded S-complex over Z:

r

C˚pY, Kq “ C˚pY, Kq ‘ C˚´1pY, Kq ‘ Z

The S-chain homotopy type of this is an invariant of pY, Kq. Its homology was identified in [DS, Theorem 8.9] with Kronheimer and Mrowka’s I˚6pY, Kq from [KM11a], up to a grading shift. The homology of C˚pY, Kq is denoted I˚pY, Kq and is called the irreducible

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instanton homology of the knot, as its chain complex is generated by (perturbed) irreducible SU p2q flat connections on Y zK with holonomy of order 4 around small meridians. We have

χ pI˚pY, Kqq “ 1

2σpY, Kq ` 4λpY q

where λpY q is the Casson invariant of Y , a result essentially due to Herald [Her97]. The S-complex structure of rC˚pY, Kq is closely tied to the following important fact. For knots in integer homology 3-spheres pY, Kq, among singular flat SU p2q connections mod gauge on pY, Kq as above, all are irreducible except for a unique reducible, which is typically denoted θ. This reducible has gauge stabilizer S1and corresponds to the abelian holonomy representation π1pY zKq Ñ SU p2q with image t˘1, ˘iu, and sends a meridian to i. The meridian is around a basepoint p P K, and its orientation is fixed by choosing an orientation of K. However the choices of the basepoint p and the orientation of K do not affect the isomorphism classes of the invariants we consider, and so they are usually suppressed from the notation.

The Frøyshov invariant associated to the S-complex rC˚pY, Kq is denoted hZpY, Kq P Z. This induces a homomorphism from the homology concordance group to the integers. The same constructions carry through for any coefficient ring R which is an integral domain. Some computations of hZpY, Kq were given in [DS, Section 9].

Motivated by constructions from [KM11b], the authors also defined versions of the above invariants with local coefficients. Generally, for an integral domain algebraS over ZrU˘1_{, T}˘1_{s, we construct a Z{4-graded S-complex over}S , denoted

r

C˚pY, K; ∆Sq, (2.5)

whose S-chain homotopy type is an invariant of pY, Kq. The variable U encodes information about the Chern-Simons functional, while T has to do with the monopole numbers of instantons. In fact there is more structure on (2.5) than just the S-chain homotopy type; see Subsection 2.6. From the S-complex rC˚pY, K; ∆Sq we obtain a Frøyshov invariant

h_SpY, Kq P Z,

also a homomorphism from the homology concordance group to Z.

The Z{4-graded irreducible homology I˚pY, K; ∆Sq is related to the Frøyshov invariant h_SpY, Kq. If h_SpY, Kq ě 0, then Proposition 2.4 directly implies:

rkpI1pY, K; ∆Sqq ě R h_SpY, Kq 2 V , rkpI3pY, K; ∆Sqq ě Z h_SpY, Kq 2 ^ . (2.6) Inequalities for the case h_SpY, Kq ď 0 are then obtained from the general properties rkpIip´Y, ´K; ∆Sqq “ rkpI3´˚pY, K; ∆Sqq and hSp´Y, ´Kq “ ´hSpY, Kq.

Given two pairs pY, Kq and pY1_{, K}1_{q of knots in integer homology 3-spheres, we may} form their connected sum pY #Y1_{, K#K}1

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Theorem 2.7 (Theorem 6.1 of [DS]). There is a chain homotopy equivalence r

C˚pY #Y1, K#K1; ∆Sq » rC˚pY, K; ∆Sq b rC˚pY1, K1; ∆Sq

ofZ{4-graded S-complexes, and it is natural with respect to split cobordisms, when defined. This result was a key ingredient in [DS] for relating the equivariant theory to the invariants defined previously by Kronheimer and Mrowka.

To pY, Kq we may also associate three equivariant Floer groups, similar in flavor to the Heegaard Floer package for knots [OS04,Ras03b], which comes with three Z{4-graded S rxs-modules fitting into an exact triangle:

¨ ¨ ¨ ÝÑ pIpY, K; ∆_SqÝÑ IpY, K; ∆i _SqÑ qÝp IpY, K; ∆_SqÝÑ pj IpY, K; ∆_Sq ÝÑ ¨ ¨ ¨ (2.8) In fact, this is an entirely algebraic construction, depending only on the associated S-complex for pY, Kq; for any S-complex one has a similar set of equivariant homology groups. See [DS, Section 4] for details. We also recall the definitions of these homology groups in Section 6.

2.3 Reducibles on negative definite cobordisms Let pW, Sq : pY, Kq Ñ pY1_{, K}1

q be a cobordism of pairs between oriented integer homology 3-spheres with knots. In particular, W is an oriented cobordism Y Ñ Y1_{, and we assume that} S is a connected, oriented surface cobordism K Ñ K1_{embedded in W . Fix a cohomology} class c P H2pW ; Zq. Let E be a U p2q-bundle over W with c1pEq “ c.

Let W`_{(resp. S}`_{) be obtained from W (resp. S) by attaching cylindrical ends to the} boundary. A singular connection associated to pW, S, cq is a connection on E|W`_zS` which

is asymptotic to connections on Y zK and Y1

zK1 with a controlled singularity along S`_. The latter condition is that the holonomy of A around a meridian of S`_{is asymptotic to}

„ i 0 0 ´i



as the size of the meridian goes to zero. The adjoint action of U p2q on sup2q, the Lie algebra of SU p2q, induces the adjoint map ad : U p2q Ñ SOp3q. We focus on singular connections A such that detpAq is a fixed (non-singular) U p1q connection λ on detpEq. Determinant one automorphisms of the bundle E which are compatible with the asymptotic and singularity conditions of singular connections define a gauge group acting on the space of singular connections. A connection is irreducible if its stabilizer with respect to this gauge group is ˘1, and is reducible otherwise. (See [KM11b] for more details on singular connections.)

The topological energy, or action, of a singular connection A is given by κpAq “ 1

8π2 ż

W`_zS`

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The curvature of adpAq extends to the singular locus and its restriction to S`_{has the form} „

Ω 0

0 ´Ω 

where Ω is a 2-form with values in the orientation bundle of S`_{. The monopole number of a} singular connection A is given by

νpAq “ i π ż S` Ω ´ 1 2S ¨ S. (2.10)

Note that our convention here is slightly different from [DS] where the self-intersection term in (2.10) does not appear. Equip W`_{with a Riemannian metric which is singular of} cone angle π along S`_{and agrees with product metrics at the ends. A singular instanton} associated to pW, S, cq is a singular connection A such that adpAq is a finite-energy anti-self-dual connection.

We write M pW, S, c; α, α1_q

dfor the moduli space, of expected dimension d, of singular instantons on pW, S, cq which are asymptotic to flat connections α on Y zK and α1 _on Y1

zK1. In general, perturbations are required to achieve transversality for these moduli spaces. We write M pα, α1_q

din the case that pW, S, cq is the product r0, 1s ˆ pY, Kq, and c “ 0, with product metric structure. This moduli space has a natural R-action and we write

˘ M pα, α1

qd´1for the quotient of the part where R acts freely.

We now discuss reducible singular instantons, or simply reducibles for short. Assume b1pW q “ b`pW q “ 0. In particular, we may assume for simplicity that λ has harmonic anti-self-dual L2curvature. Consider singular instantons on E Ñ W`

zS`that are asymptotic to reducible flat singular connections associated to pY, Kq and pY1_{, K}1_{q. The reducibles among} these singular instantons are in bijection with elements of H2pW ; Zq; the correspondence sends a singular instanton compatible with a splitting

E “ L ‘ L˚

b detpEq (2.11)

to c1pLq P H2pW ; Zq. In fact, there is a unique singular instanton ALof the form λL‘ λ˚

Lb λ, compatible with (2.11), where λLis a U p1q connection on L Ñ W`zS`with harmonic anti-self-dual L2curvature representing the cohomology class c1pLq ` 1₄S, and for which λLhas holonomy of a meridian asymptotic to i P U p1q. We will slightly abuse notation and write S for both the homology class induced by S and its Poincaré dual. Unlike in the non-singular setup, the order of the line bundle factors in the reduction (2.11) matters.

The topological energy of the reducible instanton ALis computed from (2.9) to be κpALq “ ´ ˆ c1pLq ` 1 4S ´ 1 2c ˙2 . (2.12)

The monopole number of ALis computed from (2.10) to be

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The index of the reducible ALis computed in [DS, Lemma 2.6] as follows: indpALq “ 8κpALq´ 3 2pχpW q`σpW qq`χpSq` 1 2S¨S`σpY, Kq´σpY 1_{, K}1 q´1. (2.14) Let us call AL (and c1pLq) minimal (with respect to c) if it minimizes κpALq among all reducibles on E. Define κminpW, S, cq to be κpALq for any minimal reducible AL:

κminpW, S, cq :“ min zPH2_{pW ;Zq}´ ˆ z `1 4S ´ 1 2c ˙2 . (2.15)

Note that this quantity depends only on the cohomology ring of W , the homology class of S, and the index 2 coset of c. The signed count of minimal reducibles with powers of T keeping track of monopole numbers is:

ηpW, S, cq :“ÿp´1qc1pLq2_TνpALq_{P ZrT}˘1_s _(2.16)

Here the sum is over all c1pLq corresponding to a minimal reducible ALwith respect to c. The signs are derived from [KM95, Appendix 1(ii)]. If c “ 0, we simply write κminpW, Sq and ηpW, Sq for the above quantities.

2.4 Morphisms from cobordisms

We now slightly generalize the setup from [DS], and formulate a class of cobordisms amenable for inducing morphisms between S-complexes for knots.

Definition 2.17. Let pW, Sq : pY, Kq Ñ pY1_{, K}1_{q be a cobordism of pairs between oriented} knots in integer homology 3-spheres, where S is connected and oriented, and c P H2pW ; Zq. LetS be an algebra over ZrU˘1_{, T}˘1

s. We say pW, S, cq is negative definite overS if: (i) b1pW q “ b`pW q “ 0;

(ii) the index of one (and thus every) minimal reducible ALis ´1; (iii) ηpW, S, cq is non-zero as an element inS .

If c “ 0, we also say pW, Sq is a negative definite pair overS .

Remark2.18. More general types of cobordisms will be considered in Subsection 4.2. Remark2.19. Using (2.14), condition (ii) simplifies to the following condition:

8κminpW, S, cq ` χpSq ` 1

2S ¨ S ` σpY, Kq ´ σpY 1_{, K}1

q “ 0.

Remark 2.20. Let T Ă H2pW ; Zq be the torsion subgroup of cardinality |T |. Call z P H2_{pW ; Zq{T minimal (with respect to c) if it lifts to a minimal class in H}2_{pW ; Zq. Then}

ηpW, S, cq “ |T | ¨ ÿ zPH2_{pW ;Zq{T}

minimal

p´1qz2Tp2z´cq¨S.

Consequently, if the order of torsion |T | is zero inS , then pW, S, cq is not negative definite overS .

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Suppose pW, S, cq is negative definite overS . Then we can associate to it a morphism of S-complexes, in the sense discussed in Subsection 2.1:

r

λ_pW,S,cq: rC˚pY, K; ∆Sq Ñ rC˚pY1, K1; ∆Sq (2.21) To explain this we recount some more background on the construction of rC˚pY, K; ∆Sq.

First, we mention the two key identities on which the local coefficients structure relies. First, let pW, Sq : pY, Kq Ñ pY1_{, K}1

q be a cobordism of pairs with S oriented, and suppose α and α1 _{are singular SU p2q connections on Y zK and Y}1_zK1 _{respectively. Let E be a} U p2q-bundle over W with c “ c1pEq. Then if A is a singular connection on W zS which restricts to α and α1_{at the boundary components, we have:}

2κpAq ´ 2κminpW, S, cq ” CSpαq ´ CSpα1q pmod Zq. (2.22) Furthermore, we note that 2κminpW, S, cq ” ´1₈pS ´ 2cq2 pmod Zq. The above may be taken as a definition of CSpαq P R{Z by letting α1_{be the reducible. The second identity is:} νpAq ” holK1pα1q ´ hol_Kpαq pmod Zq. (2.23)

Here holKpαq is the limiting U p1q holonomy around a Seifert longitude for K of the restriction of α to the complex line bundle L defining the reduction near K.

We next recall from [DS, Subsection 7.1] that C˚pY, K; ∆Sq “ À

α∆αwhere the sum is over (perturbed) irreducible flat SU p2q connections α on Y zK with traceless holonomy around meridians, and ∆αis defined to be theS -module

∆α:“S ¨ U´CSpαqTholKpαq.

Here CSpαq is any lift to R of the Chern–Simons invariant of α, and similarly for holKpαq. The maps d, v, δ1, δ2 for C˚pY, K; ∆Sq are defined by certain counts of singular instantons rAs on R ˆ Y , where each instanton has weight U2κpAqTνpAq. For example, we have

dpαq “ÿεpAq∆pAqpαq

where the sum is over singular instantons rAs on R ˆ Y in 0-dimensional moduli spaces ˘

M pα, α1_q

0and ∆pAq : ∆α Ñ ∆α1 is the module homomorphism which is multiplication

by U2κpAqTνpAq. The sign εpAq “ ˘1 is determined by the orientation of the moduli space. When carrying out these constructions generic metrics and perturbations are chosen, but we typically omit these from our discussions. For more details see [DS].

We now assume pW, S, cq is negative definite over S . To construct the morphism r

λpW,S,cqwe must prescribe the maps appearing in the decomposition (2.1). The components λ, µ, ∆1, ∆2 are defined just as in [DS, Section 3], adapted to the local coefficients setting of [DS, Section 7]. For example, λpαq “ ř εpAq∆pAqpαq where the sum is over rAs P M pW, S, c; α, α1q0and ∆pAq : ∆αÑ ∆α1 is multiplication by:

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The other components of rλ_pW,S,cqare defined similarly. The term η PS in the decomposition (2.1) is defined to be the count of reducibles ηpW, S, cq from (2.16), which is non-zero by assumption. The compatibility of these maps with the local coefficient systems is owed to (2.22) and (2.23).

Having prescribed the components of rλ_pW,S,cq, the relations λd ´ d1_{λ “ 0,} _{µd ` λv ` ∆}

2δ1´ v1λ ` d1µ ´ δ21∆1“ 0

follow just as for negative definite pairs; see e.g. [DS, Proposition 3.19]. The relations ∆1d ` ηδ1´ δ11λ “ 0, d1∆2´ ηδ21 ` λδ2 “ 0

are only slight modifications of those in [DS, Proposition 3.10]; in the argument, one must keep track of trajectory breakings at all possible minimal reducibles with index ´1, instead of a unique flat reducible. There is one technical point to make here: unlike for a negative definite pair, a general pW, S, cq which is negative definite overS may a priori have obstructed (degenerate) minimal reducibles. However, this can be rectified by using a small perturbation, for example as is done in [DCX17, Section 7.3]. We obtain:

Proposition 2.24. Let pW, S, cq : pY, Kq Ñ pY1_{, K}1

q be negative definite overS . Then there is an induced morphism ofS-complexes rλ_pW,S,cq: rC˚pY, K; ∆Sq Ñ rC˚pY1, K1; ∆Sq. Remark2.25. Suppose c “ 0, H1pW ; Zq “ 0, and the homology class of S is divisible by 4. The minimum (2.15) is achieved uniquely by z “ ´1₄S. Call the corresponding reducible AL. Then κminpW, Lq “ κpALq “ 0 and νpALq “ ´1₂S ¨ S. In particular, ALis flat, and ηpW, Sq “ T´1₂S¨S

‰ 0. Thus pW, Sq is negative definite overS , for any S . This is the class of “negative definite pairs” in [DS, Defintion 2.33].

Remark2.26. If we drop the condition (iii) in Definition 2.17, so that ηpW, S, cq may be zero, then we still have a map rλpW,S,cq which respects the structure of the S-complexes. However, it is not a morphism in the terminology of Subsection 2.3.

Remark 2.27. In defining the cobordism map for pW, S, cq, a path on S that interpolates between chosen basepoints on K and K1 _{must be specified to define the component µ.} However, this choice is never important for us and so we suppress it from the notation. Remark2.28. A more functorial discussion for cobordism maps replaces c P H2pW ; Zq with an oriented 2-manifold properly embedded in W transverse to S which represents the Poincaré dual to c. This is because such a submanifold naturally determines a U p2q bundle over pW, Sq, compare [KM11a]. However, this is not important for our purposes and in this paper we settle for using cohomology classes c P H2pW ; Zq.

2.5 Some cobordism relations

The importance of using local coefficients and in particular the invertibility of T4´ 1 comes from the effect, on the singular instanton invariants, of some basic moves on surfaces in

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4-manifolds. These relations were first established in the closed case by Kronheimer [Kro97], and have played an important role in Kronheimer and Mrowka’s work on singular instanton Floer homology; see [KM11b,KM13,KM19a,KM19b]. Here we state the relations as they apply in our current setup, at the chain level. The proof is a direct adaptation of the one for [Kro97, Proposition 3.1]; see also [KM11b, Proposition 5.2].

Thus far, our cobordisms of pairs pW, Sq : pY, Kq Ñ pY1_{, K}1

q have assumed the surface S to be embedded in W . We slightly relax our above hypotheses so that S may be a smoothly immersedcobordism with normal crossings, i.e. transverse double points. Blow up W at each double point of S to obtain W , diffeomorphic to a connected sum of W with copies of CP2, and let S Ă W be the proper transform of S. We say pW, S, cq is negative definite overS if pW , S, cq satisfies the conditions of Definition 2.17. In this case we define

r

λpW,S,cq:“ p´1qs`T´2s`rλ

pW ,S,cq. (2.29)

Here s`is the number of positive double points of S. The factor p´1qs`T´2s` is not so important and is only included to symmetrize some expressions. Here c P H2pW ; Zq, and we write c also for its image under H2pW ; Zq Ñ H2pW ; Zq.

Proposition 2.30. Suppose pW, S, cq is negative definite over S , where S is possibly immersed inW as above. Let S˚_{be obtained from}_{S by either a positive twist move or a} finger move, which introduces a cancelling pair of double points. Then

r

λpW,S˚_,cq „ pT2´ T´2qrλ_pW,S,cq

where „ stands for S-chain homotopic, as morphisms of Z{4-graded S-complexes overS . IfS˚_{is obtained by a negative twist move, then r}_λ

pW,S˚_,cqisS-chain homotopic to rλ_pW,S,cq.

Proof. As already mentioned, the proof is adapted from [Kro97] to our current setup. For this reason we omit some of the details and only highlight the key ingredients.

Without loss of generality, we may assume S is smoothly embedded in W , and c “ 0. Suppose S˚ _{is obtained from S by a positive twist move. Then the map r}_λ

pW,S˚_q “

´T´2rλ

pW ,S˚qis defined by choosing a metric and generic perturbation for the pair pW , S˚q “ pW, Sq#pCP2, S2q (2.31) where S2is a conic in CP

2

, and in particular rS2s “ 2e where e P H2pCP 2

; Zq is a generator. Thus S2¨ S2 “ ´4. Consider reducibles on pCP

2

, S2q. There are exactly two that have minimal topological energy; the two reducibles A0and A´ecorrespond, respectively, to the minimizers z “ 0 and z “ ´e in (2.12), in which we set c “ 0. Thus κmin“ 1{4.

A 1-parameter family of auxiliary data that stretches along the connected sum sphere induces an S-chain homotopy between rλ

pW ,S˚qand the morphism rλ

8_{for the pair (2.31)} defined using a broken metric along the connected sum region. A gluing argument shows

r

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where rλ_pW1_,S1_qis the morphism for pW1, S1q, a puncturing of pW, Sq, equipped with auxiliary

data involving a metric with cylindrical ends at the punctures. In other words, the instantons that contribute to rλ8 _{can be described as instantons on pW}1_{, S}1

q grafted to the minimal reducible instantons on pCP2, S2q, which are unobstructed. For the two minimal reducibles A0and A´ewe compute the monopole numbers:

νpAzq “ 2z ¨ 2e “ #

0, z “ 0 4, z “ ´e Thus ηpCP2, S2q “ 1 ´ T4.

Finally, a 1-parameter family of auxiliary data from any given metric and perturbation that defines rλpW,Sq and which stretches along the 3-sphere which encloses the connected sum location on S provides a chain homotopy from rλpW,Sqto the morphism rλpW1_,S1_q.

The other moves are dealt with by similarly adapting [Kro97].

Corollary 2.32. Suppose pW, S, cq is negative definite over S , where the surface S is possibly immersed inW with transverse double points.

(i) IfS˚_{is obtained from}_{S by a negative twist, then pW, S}˚_{, cq negative definite over}S . (ii) If T4 ‰ 1 and S˚ is obtained from S by a positive twist or a finger move, then

pW, S˚, cq is negative definite overS .

2.6 Chern–Simons filtration and enriched complexes

In this subsection we review the structure of the Chern–Simons filtration on our S-complexes for knots. We will be brief, and refer to [DS, Section 7] for more details.

Definition 2.33. An I-graded S-complex over RrU˘1_{s is an S-complex p r}_{C, r}_{d, χq over} RrU˘1

s with a Z ˆ R-bigrading as an R-module which satisfies the following. Write rCi,j for the pi, jq P Z ˆ R graded summand. Then

(i) U rCi,j Ă rCi`4,j`1 (ii) rd rCi,j Ă

Ť

kăjCri´1,k (iii) χ rCi,j Ă rCi`1,j

Also, rC is generated as an RrU˘1_{s-module by homogeneously bigraded elements. The} distinguished summand in rC, isomorphic to RrU˘1

s, has generator 1 in bigrading p0, 0q. For an I-graded S-complex we writegr for the Z-grading and degr Ifor the R-grading, which we also call the “instanton” grading. When we write rC˚for an I-graded S-complex, the subscript only records the Z-grading, or the induced Z{4-grading.

Note that if we forget the R-grading, we still have a Z-graded S-complex in the sense that rC˚is a graded module over RrU˘1s, where the latter is a graded ring with U in degree

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4. However, in this paper, for simplicity the algebraic objects we consider are either I-graded S-complexes as above, or Z{4-graded S-complexes as discussed earlier.

Suppose no perturbations are needed when constructing an S-complex for pY, Kq. Then rCpY, K; ∆_Sq has the structure of an I-graded S-complex withS “ RrU˘1s and R “ ZrT˘1_{s, as we now explain. For a flat singular connection α on Y zK choose a path to} the reducible, i.e. a singular SU p2q connection on R ˆ Y with limits α and θ at ´8 and `8, respectively. Call the gauge equivalence class of this connectionα. We call such ar choice ofα a lift of α. The Chern–Simons invariant ofr α isr

CSpαq :“ 2κpr αq P R,r

and is a lift of CSpαq P R{Z. The invariant CSpαq depends only on the homotopy class ofr r

α, viewed as a path in the configuration space of singular connections on R ˆ Y relative to its limits α and θ. Similarly, the pathα determines a lift holr Kpαq P R of the holonomyr invariant holKpαq P R{Z.

Given a singular connection A on R ˆ Y , we can “add an instanton” by grafting on to A a standard instanton on S4. This changes pκpAq, νpAqq into pκpAq ` 1, νpAqq. Similarly, we can “add a monopole” by grafting on to A a standard singular instanton on pS4, S2q. This changes pκpAq, νpAqq to pκpAq ` 1{2, νpAq ´ 2q. In the case that A represents a lift

r

α of α, we can faithfully generate the set of all lifts of α by a sequence of these operations and their inverses. The pair pCS, holKq defines a bijection from the set of lifts of α into a subspace of R2which is given as

pCSprα0q, holKprα0qq ` tpi, 2jq P Z2 | i ” j mod 2u,

withαr0being a fixed lift of α. The above shows that the correspondence defined by r

α ÞÑ UCSprαq´CSprα0q_TholKprαq´holKprα0q

r

α0 (2.34)

allows us to identify an index 4 subgroup of the monomials in ZrU˘1_{, T}˘1_sr_α

0with the lifts of α. Motivated by this, we make the following.

Definition 2.35. An honest lift of a (perturbed) flat connection α is a choice of pathα ofr connections mod gauge from α to the reducible θ. We consider honest lifts up to homotopy relative to the endpoints. A lift of α is a monomial UiTjα wherer α is an honest lift. The setr of all such lifts are denoted byPα.

We extend pCS, holKq to any lift of α by setting CSpUiTjαq “ i ` CSpr αq,r holKpU

i_Tj r

αq “ j ` holKpαq.r (2.36) Turning to the I-graded S-complex structure, the instanton gradings are defined by

deg_Iprαq :“ CSprαq P R.

and we extend this to rC˚pY, K; ∆Sq using (2.36). The index of the ASD operator forαr determinesgrpr αq P Z, which in this case is the expected dimension of the moduli spacer

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of instantons in the same homotopy class as α. We extend the Z-gradingr gr to all ofr r

C˚pY, K; ∆Sq by settinggrpUr i_Tj

r

αq “ 4i `grpr αq. The reducible θ is identified with ther constant path rθ, with bigrading p0, 0q. The component of α1_{in dp}

r

αq is given by the following sum over rAs P ˘M pα, α1

q0:

ÿ

εpAq∆pAqpαqr

where ∆pAq : Pα Ñ Pα1 is defined by requiring that the concatenation of A withα_r1

is homotopic to the pathα. Fixing a lift for each generator of rr C˚pY, K; ∆Sq and using (2.34) we can regard the above expression as an element of ZrU˘1_{, T}˘1_{s. The other maps} are described similarly. That rC˚pY, K; ∆Sq is an I-graded S-complex follows from the additivity of topological energy, which implies

2κpAq “ CSpαq ´ CSpr αr 1

q

in this situation, and the fact that the energy κpAq is always non-negative for instantons, and is positive for instantons on the cylinder with α ‰ α1_.

A morphism rλ : rC˚ Ñ C˚1 of levelδ P R of I-graded complexes is an RrU˘1s-module homomorphism which is a morphism of S-complexes and satisfies

r λ rCi,j Ă ď kďj`δ r C_i,k1 (2.37)

When δ “ 0 we simply call rλ a morphism. Similarly, a chain homotopy of level δ of I-graded S-complexes is a chain homotopy of the underlying S-complexes which increases the instanton grading by at most δ. We may also form tensor products, duals, and local equivalence classes of I-graded S-complexes.

Proposition 2.38. Let pW, S, cq : pY, Kq Ñ pY1_{, K}1_{q be negative definite over}S . Suppose no perturbations are required in the construction of rλpW,S,cqfrom Proposition 2.24. Then this map lifts to a morphism of level2κminpW, S, cq of I-graded S-complexes.

The lift is straightforward to define, following the construction of Proposition 2.24. For in-stance, the contribution of α1_{in λpr}_{αq may be defined as a sum over rAs P M pW, S, c; α, α}1_q

0: ÿ

εpAq∆pAqpαqr

where ∆pAq :Pα ÑPα1 maps a liftα of α to the lift_r α_r1of α1 satisfying

2κpAq ´ 2κminpW, S, cq “ CSprαq ´ CSpαr 1

q, νpAq “ holK1pα1q ´ hol_Kpαq. (2.39)

The other components are described similarly. That the resulting morphism is of level 2κminpW, S, cq follows from the first identity in (2.39). The above definitions can be adapted to the case that R is an algebra over the ring ZrT˘1_{s in a straightforward way.}

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Lemma 2.40. Suppose α and α1_{are generators of}_C

˚pY, K; ∆Sq with honest liftsα andr r

α1_{. Then the matrix entry xdpr}_αq, r

α1_{y has the form} UmT2mÿ iPZ aiT4iP ZrU˘1, T˘1s (2.41) wherem “ pgrpr αr 1_{q ´} r

grpαq ´ 1q{4. A similar result holds for other matrix entries of rr d and for a morphism rλpW,S,cqassociated to a negative definite pair; that is to say, any matrix entry is a Laurent polynomial of the above form for an appropriate choice ofm.

In general we cannot get by without using perturbations. In the terminology of [DS], the I-graded S-complexes of Definition 2.33 are of “level 0”. More generally, an I-graded complex of level δ ě 0 is defined as in Definition 2.33, except that in (ii) we allow rd to increase the instanton grading by at most δ. Then rCpY, K; ∆_Sq always has the structure of an I-gradedS -complex of level some small δ ě 0 determined by the perturbation data. (Alternatively, one can always give it the structure of an I-graded complex, with the caveat that instanton gradings are tethered to a perturbed Chern–Simons functional.)

An enriched complex is a sequence of I-graded S-complexes of levels δi ě 0 with limiÑ8δi “ 0, where the complexes are related to one another by suitable morphisms. Associated to a sequence of admissible perturbation data going to zero is an enriched complex for pY, Kq, unique up to chain homotopy of enriched complexes. All invariants of pY, Kq considered here may be derived from it. For more details see [DS, Section 7]. Remark2.42. In this paper we typically work with I-graded S-complexes instead of enriched complexes. In fact, most of the examples in our computations require no perturbations, and so do not require the formalism of enriched complexes. The remaining computations involve the connected sum of knots such that each of the summands does not require any perturbation. Theorem 2.7 for connected sums holds at the level of enriched complexes, see [DS, Theorem 7.20]. So for the knots that appear in our computations this theorem provides provides I-graded S-complexes.

2.7 The concordance invariant ΓK

We next review the definition of concordance invariants ΓR_pY,Kq from [DS]. These invari-ants, similar to the case of the Frøyshov invariinvari-ants, are obtained by applying an algebraic construction to the enriched complex for pY, Kq.

We first define the invariant Γ_C_rfor an I-graded S-complex rC “ C˚‘ C˚´1‘ RrU˘1s over RrU˘1

s. For α “ř skζkP C where ζkare homogeneous of distinct bigradings and skP R, define degIpαq to be the maximum of the degIpζkq such that sk‰ 0. The invariant Γ

r

C is a function from the integers to Rě0:“ Rě0Y 8. For k a positive integer, set Γ

r

Cpkq :“ inf pdegIpαqq P Rě0 where the infimum is over α P C withgrpαq “ 2k ´ 1 such thatr

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For k a non-negative integer we define

Γ_C_rpkq :“ max pinf pdegIpαqq , 0q P Rě0

where the infimum is over α P C with grpαq “ 2k ´ 1 such that there exist elementsr a0, . . . , a´k P RrU˘1s with a´k ‰ 0 and

dα “ ´k ÿ i“0 viδ2paiq. Let S “ RrU˘1

s where R is an algebra over ZrT˘1s. If rC˚pY, K; ∆Sq has the structure of an I-graded S-complex, then we define

ΓR_pY,Kq:“ Γ_C_r

˚pY,K;∆Sq

In the more general case, for an enriched complex, one should take an appropriate limit of the above construction. We refer to [DS, Section 7] for details.

When R “ ZrT˘1_{s we omit it from the notation, and write Γ}

pY,Kq. Similarly, when K Ă S3we write ΓR_K. For basic properties of these invariants see [DS, Theorem 7.24].

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3 Computations for 2-bridge knots

In this section, we firstly recall some of the computations of [DS, Section 9] where we used the ADHM description of the moduli of instantons on R4to study the S-complexes of 2-bridge knots. There we characterized the S-complex of the trefoil. In Subsections 3.2 and 3.3, respectively, we consider two families of 2-bridge knots both of which generalize the trefoil example: 2-bridge torus knots and double twist knots. In Subsection 3.4 we compute Γ-invariants for connected sums of double twist knots. Finally, in Subsection 3.5 we compute the Γ-invariants for some 2-bridge knots with 11 crossings.

3.1 Background

Let Kp,qdenote the 2-bridge knot whose branched double cover is the lens space Lpp, qq with p being an odd integer. The S-complex of Kp,qwas studied in [DS, Section 9], which we recall here. There are pp ` 1q{2 singular flat SU p2q connections on S3zKp,qup to gauge, denoted ξ0, ξ1, . . . , ξpp´1q{2_{where ξ}0

“ θ is the unique reducible. LetS “ ZrT˘1s. Then the underlying module of the S-complex rC˚“ rC˚pKp,q; ∆Sq of Kp,qis given by

r C˚“ C˚‘ C˚´1‘S , C˚“ p´1 2 à i“1 S ¨ ζi

where ζi is a lift of ξi. The remaining part of the S-complex structure on rC˚is given by the maps d, v : C˚ Ñ C˚, δ1 : C˚ Ñ S , δ2 : S Ñ C˚, which are defined using the unparametrized moduli spaces ˘M pξi, ξjqdwith dimension d ď 2.

All of the critical points and moduli spaces here are non-degenerate. Given an instanton rAs P ˘M pξi, ξjqd, there are positive integers k1, k2and ε1, ε2P t˘1u satisfying

k1 ” ε1i ` ε2j pmod pq, qk2 ” ´ε1i ` ε2j pmod pq (3.1) and which furthermore satisfy the following relations:

d “ N1pk1, k2; p, qq ` 1

2N2pk1, k2; p, qq ´ 1, 2κpAq “ k1k2

p . (3.2)

Here N1pk1, k2; p, qq is the number of pa, bq P Z2with |a| ă k1, |b| ă k2solving

a ` qb ” 0 pmod pq (3.3)

and similarly N2pk1, k2; p, qq is the number of solutions pa, bq P Z2 with either |a| “ k1, |b| ă k2 or |a| ă k1, |b| “ k2. Note that pa, bq “ p0, 0q always satisfies (3.3) and given any solution pa, bq of (3.3), p´a, ´bq is also a solution. Thus N1pk1, k2; p, qq is an odd positive integer and N2pk1, k2; p, qq is an even non-negative integer.

The 0-dimensional moduli space ˘M pξi_{, ξ}j_q

0 is non-empty only if there are k1 and k2 satisfying (3.1) such that the following hold:

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In fact, if there are such k1 and k2, then ˘M pξi, ξjq0 consists of two oppositely oriented points, related to each other by the so-called flip symmetry, which is an involution acting on the moduli spaces. The set of monopole numbers for this pair of singular instantons is t2, ´2u if k1and k2are both odd and t0u otherwise.

For any ξi, we assume the lift ζi satisfies holKpζiq “ 0. This is possible because holKpξiq is zero in R{Z. (However, note that this lift might not always be honest in the sense of Definition 2.35.) Then the matrix entry xdζi, ζjy, for 1 ď i, j ď p´1₂ , is non-zero if and only if there are odd positive integers k1, k2satisfying (3.1) and (3.4). We obtain

xdζi, ζjy “ #

˘pT2´ T´2q D odd k1, k2 P Zą0solving (3.1) and (3.4)

0 otherwise (3.5)

By letting j “ 0 (resp. i “ 0) above, we obtain a similar characterization of the element δ1pζiq (resp. xδ2p1q, ζjy).

There is a similar characterization for the 1-dimensional moduli spaces ˘M pξi, ξjq1, which are the geometrical input in the definition of the map v. The space ˘M pξi, ξjq1 is non-empty if and only if there are positive integers k1 and k2satisfying (3.1) such that

N1pk1, k2; p, qq “ 1, N2pk1, k2; p, qq “ 2. (3.6) Moreover, as before, the set of monopole numbers for instantons in ˘M pξi, ξjq1is t2, ´2u if k1and k2are odd and is t0u otherwise. Thus we have

xvζi, ζjy “ T2A`` T´2A´` B where either B “ 0 or A`

“ A´“ 0 depending on whether the relevant integers k1and k2 for the moduli space ˘M pξi, ξjq1are odd or not. It is shown in [DS] that the homology of

r

C˚after evaluating T at 1 and working with coefficient ring Z is isomorphic to I6pKp,q; Zq. This is free abelian of rank p, by [KM11a, Corollary 1.6]. As rC˚has rank p, we must have A´

“ ´A`, and B “ 0. Consequently, we have:

xvζi, ζjy “ npT2´ T´2q, n P Z, and (3.7) n ‰ 0 ùñ D odd k1, k2 P Zą0solving (3.1) and (3.6)

The congruences (3.1) can be also used to determine Floer and instanton gradings. Let k1and k2be any pair of positive integers that solve (3.1) where 0 ď i, j ď p´1₂ . Then there are lifts ζiand ζj of ξiand ξj respectively such that their Z ˆ R bigradings satisfy

` r grpζiq, deg_Ipζiq˘´`grpζr j q, deg_Ipζjq˘“ pN1pk1, k2; p, qq ` 1 2N2pk1, k2; p, qq, k1k2 p q (3.8)

We always assume ζ0is the constant path for θ, so that its bigrading is p0, 0q. Having fixed ζ0, if j “ 0 (resp. i “ 0), then imposing (3.8) for some k1, k2determines the bigrading of ζi

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(resp. ζj). If instead i and j are both non-zero, it determines ζiand ζjup to simultaneously multiplying them by Ukfor some k P Z, which alters bigradings by p4k, kq. It is important to note that the rule (3.8) depends on k1, k2.

Now consider rC˚pKp,q; ∆Sq as an I-graded S-complex overS “ RrU˘1s. Suppose k1, k2 solve (3.1) and (3.4), and i, j ‰ 0. If we choose lifts ζi and ζj satisfying (3.8), then (3.5) continues to hold. Letting one of i or j be zero gives an analogous statement for δ1and δ2. Next, suppose i, j ‰ 0 and k1, k2solve (3.1) and (3.6). If in this case we choose lifts ζi and ζj satisfying (3.8), then (3.7) continues to hold. (Different choices of lifts in any of these situations might introduce powers of U into the expressions.)

Remark3.9. The authors expect that a more thorough investigation of the equivariant ADHM description of instantons on R4in this context should allow for the direct computation of the v-maps for 2-bridge knots. This would yield a complete combinatorial description of the I-graded S-complexes for 2-bridge knots, and their Γ-invariants.

3.2 The torus knots T2,2k`1

In this section we determine the full structure of the invariants defined in [DS] for the p2, 2k ` 1q torus knots T2,2k`1where k is a positive integer.

Proposition 3.10. The I-graded S-complex rC˚ “ rC˚pT2,2k`1; ∆Sq is given by

r C˚ “ C˚‘ C˚´1‘S , C˚ “ k à i“1 S ¨ ζi_. The differential rd has the components d “ δ2 “ 0 and

δ1`ζi ˘ “ # pT2´ T´2q i “ 1 0 2 ď i ď k v`ζi˘“ # pT2´ T´2qζi´1 2 ď i ď k 0 i “ 1

TheZ ˆ R bigrading of ζiis given by p2i ´ 1, i2{p2k ` 1qq.

Proof. Most of the statement follows from the content of Subsection 3.1. The torus knot T2,2k`1is the 2-bridge knot Kp,´1where p “ 2k ` 1. In this case, the congruence (3.3) is a ” b (mod p). The constraints N1pk1, k2; p, qq “ 1, N2pk1, k2; p, qq “ 0 in (3.4) imply that p0, 0q is the only solution to a ” b (mod p) with |a| ď k1, |b| ď k2, with one of these two inequalities strict. This is true if and only if k1“ k2“ 1, and from (3.1), this in turn happens if and only if j “ 0, i “ 1, and ε1 “ 1. Thus d “ δ2 “ 0, and δ1`ζi

˘

is equal to ˘pT2´ T´2q if i “ 1 and is zero otherwise. Here we have chosen the lift ζ1of ξ1satisfying (3.8), so that it has Z ˆ R bigrading p1,_p1q.

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We now turn to the map v. The equations (3.3) and (3.6) hold simultaneously if and only if one of k1, k2is equal to 1 and the other is an odd integer ą 1 and ă p. This implies that i “ j ´ 1. Choose the lifts ζiof ξi for 2 ď i ď k satisfying (3.8):

` r grpζiq, degIpζiq ˘ ´`grpζr i´1 q, degIpζi´1q ˘ “ p2,p2i ´ 1q p q. (3.11)

This implies that the Z ˆ R bigrading of ζiis given by p2i ´ 1, i2{p2k ` 1qq. In the basis ζ1, . . . , ζkthe map v has the form:

v “ pT2´ T´2q » — — — — — — – 0 a1 0 ¨ ¨ ¨ 0 0 0 a2 ... .. . . .. ak´1 0 ¨ ¨ ¨ 0 fi ffi ffi ffi ffi ffi ffi fl .

To compute the entries ai we use Theorem 7, which asserts when T2´ T´2 ‰ 0 that h_SpKp,´1q “ ´σpKp,´1q{2 “ k.

Proposition 2.4 implies there is an α P C˚ with δ1vk´1pαq ‰ 0, which happens only if a1, . . . , ak´1are nonzero. This holds over any characteristic, and so ai “ ˘1. After possibly replacing ζi with ´ζiwe may assume ai “ 1, and v has the claimed form.

Corollary 3.12. For any k, the Γ-invariant of T2,2k`1is given as

ΓT2,2k`1piq “ $ ’ & ’ % 0, i ď 0 i2 2k`1, 1 ď i ď k 8, i ě k ` 1

The case k “ 1 of Proposition 3.10 recovers the trefoil complex determined in [DS, Section 9]. This simple type of complex appears in the sequel, and so we make the following. Definition 3.13. For any positive real number t we let rCptq “ C˚ ‘ C˚´1 ‘S be the I-graded S-complex where C˚is freely generated by a single generator ζ with bigrading

pgrpζq, degr Ipζqq “ p1, tq. The differential has δ1pζq “ T2´ T´2while d, v, δ2are zero.

The trefoil T2,3has an I-graded S-complex isomorphic to rCp1₃q. We note that the Frøyshov invariant of rCptq is equal to 1 if T4 _{‰ 1 in} _{S and is otherwise zero. Recall that the} undecorated Γ-invariant for an I-graded S-complex is to be understood as defined for the coefficient ringS “ ZrT˘1

s. The Γ-invariant of the complex rCptq is given by:

Γ r Cptqpiq “ $ ’ & ’ % 0, i ď 0 t, i “ 1 8, i ě 2

Chern-Simons functional, singular instantons, and the four-dimensional clasp number

Chern-Simons functional, singular instantons, and the

four-dimensional clasp number

Contents

1

Introduction

2

Preliminaries

3

Computations for 2-bridge knots